| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,1,2,0,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1031'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833'] |
| Outer characteristic polynomial of the knot is: t^7+51t^5+72t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1031'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 192*K1**3*K2*K3 - 128*K1**3*K3 - 192*K1**2*K2**4 + 832*K1**2*K2**3 - 3264*K1**2*K2**2 - 64*K1**2*K2*K4 + 2816*K1**2*K2 - 1492*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2496*K1*K2*K3 + 80*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 952*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 792*K2**2 + 80*K2*K3*K5 - 476*K3**2 - 102*K4**2 + 1116 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1031'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16913', 'vk6.17155', 'vk6.17499', 'vk6.17507', 'vk6.17554', 'vk6.17562', 'vk6.21893', 'vk6.24023', 'vk6.24035', 'vk6.24102', 'vk6.27951', 'vk6.29430', 'vk6.35327', 'vk6.35759', 'vk6.36283', 'vk6.36293', 'vk6.36358', 'vk6.39359', 'vk6.41536', 'vk6.43436', 'vk6.43444', 'vk6.43471', 'vk6.45924', 'vk6.47613', 'vk6.55072', 'vk6.55321', 'vk6.55613', 'vk6.55621', 'vk6.55648', 'vk6.58544', 'vk6.60123', 'vk6.60135', 'vk6.60165', 'vk6.63032', 'vk6.64909', 'vk6.65122', 'vk6.65320', 'vk6.65358', 'vk6.68492', 'vk6.68519'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3O4U5U1O5U2O6U3U4U6 |
| R3 orbit | {'O1O2O3O4U5U1O5U2O6U3U4U6', 'O1O2O3O4U2U5U1O5O6U3U4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U1U2O5U3O6U4U6 |
| Gauss code of K* | O1O2O3U4U5U1U2O6O4U6O5U3 |
| Gauss code of -K* | O1O2O3U1O4U5O6O5U2U3U4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 0 2 -1 2],[ 2 0 0 1 2 2 2],[ 1 0 0 1 2 1 2],[ 0 -1 -1 0 1 0 2],[-2 -2 -2 -1 0 -2 1],[ 1 -2 -1 0 2 0 2],[-2 -2 -2 -2 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -2 -2 -2],[-2 -1 0 -2 -2 -2 -2],[ 0 1 2 0 0 -1 -1],[ 1 2 2 0 0 -1 -2],[ 1 2 2 1 1 0 0],[ 2 2 2 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,1,1,2,-1,1,2,2,2,2,2,2,2,0,1,1,1,2,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,1,2,0,1,1,1,1,-1] |
| Phi of -K | [-2,-1,-1,0,2,2,-1,1,1,2,2,1,1,1,1,0,1,1,0,1,1] |
| Phi of K* | [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,1,2,0,1,1,1,1,-1] |
| Phi of -K* | [-2,-1,-1,0,2,2,0,2,1,2,2,1,1,2,2,0,2,2,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+37t^4+25t^2+1 |
| Outer characteristic polynomial | t^7+51t^5+72t^3+4t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 192*K1**3*K2*K3 - 128*K1**3*K3 - 192*K1**2*K2**4 + 832*K1**2*K2**3 - 3264*K1**2*K2**2 - 64*K1**2*K2*K4 + 2816*K1**2*K2 - 1492*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2496*K1*K2*K3 + 80*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 952*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 792*K2**2 + 80*K2*K3*K5 - 476*K3**2 - 102*K4**2 + 1116 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{3, 6}, {5}, {4}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |